 myself, Sunil Kalshetti, assistant professor, department of electronics engineering, Walshan institute of technology, Solaapur. Today, I am going to discuss the PI and PID controllers, learning.com. At the end of this session, students can analyze PI and PID controllers, PI controller, proportional integral controllers, a controller in the forward path, which changes the controller output corresponding to the proportional plus integral of the error signal is called as a PI controller, is called as a proportional integral controller. The proportional integral controller produces an output, which is the combination of outputs of the proportional and integral controllers. Let us derive the transfer function of PI controller. Here u of t is the actuating signal and e of t is the error signal. So, u of t is equal to k p into e of t plus k i integration of e of t d t, apply the Laplace transform on both side. Therefore, u of s is equal to k p plus k i upon s into e of s. Here, u of s is the Laplace of u of t and e of s is the Laplace of e of t. Therefore, u of s upon e of s is nothing but transfer function. As we know, the transfer function is the ratio of Laplace of output to the Laplace of input is equal to k p plus k i upon s. So, this is the transfer function for the PI controller. Now, the transfer function of the PI controller is k p plus k i of s and the block diagram is as shown in the figure. So, this is the block diagram of the PI controller. Assuming k p is equal to 1, so we can write g of s is equal to 1 plus k i upon s into omega n square divided by s into s plus 2 zeta omega n. So, it is equal to k i plus s into omega n square divided by s square into s plus 2 zeta omega n. So, the system becomes type 2 in nature. So, c of s upon r of s is equal to 1 upon 1 plus g of s h of s. So, it is equal to k i plus s into omega n square divided by s cube plus 2 zeta omega n square plus s omega n square plus k i into omega n square. That is, it becomes the third order. Now, as order increases by 1, the system relatively becomes less stable as k i must be designed in such a way that the system will remain in stable condition. The second order system is always a stable system. So, k p is equal to limit s tends to 0, g of s h of s is equal to infinity. So, steady state error is equal to 0 and k v is equal to limit s tends to 0 as g of s h of s is equal to infinity. So, steady state error is equal to 0. Through the p i controller, we are adding one pole at the origin and one 0 somewhere away from the origin. As the pole is at origin, its effect will be more. Hence, p i controller may reduce the stability, but its main advantage is that it reduces the steady state error drastically. Due to this reason, it is the one of the most widely used controllers in commercial applications. Now, effect of p i controller, it increases the order of the system. It increases the type of system. Design of k i must be proper to maintain the stability of the system. So, it becomes system relatively less stable. Steady state error reduces tremendously for same type of input. Let us see p i d controller. The proportional integral derivative controller produces an output, which is the combination of the output of proportional integral and derivative controller. So, it forms the p i d controller. A p i d controller is generally used in industrial control application to regulate the stability at the temperature, flow, pressure, speed and other process variables. As the proportional derivative improves the transients and the proportional integral controller improves the steady state, then the combination of these two may be used to improve the overall time response of the p i d controller working principle. Integral tuning attempts to remedy this by effectively calculating the error results from the p action to increase the correction factor. For example, if the ohm remained below temperature, then I would act to increase the heat delivered. However, rather than stop heating, when the target is reached, I attempt to drive the cumulative error to 0 resulting in an overshoot, derivative tuning. Attempts to minimize this overshoot by showing by slowing the correction factor applied as the target is approached. So, this is the block diagram of p i d controller. Now derive the transfer function of p i d controller. So, u of t is equal to k p into e of t plus k i integration of e of t d t plus k d derivative of e of t d t. Now, take the Laplace of both side. So, u of s is equal to k p plus k i upon s plus k d s into e of s. So, u of s upon e of s is equal to k p plus k i upon s plus k d into s. Therefore, the transfer function of the proportional integral derivative controller is k p plus k i upon s plus k d into s. Again simplify this. Now it can be observed that one pole at origin is fixed and the remaining parameter k d, k p and k i decides the position of the two zeros. In this case we can keep two complex zeros or two real zeros as per the requirement. Hence p i d controller can provide better tuning. In the olden days the p i controller was one of the best choice of the control engineers because designing of p i d controller was little difficult. But towards due to the development of the software designing of p i d controller have become an easy task. The p i d controller is used to improve the stability of the control system and to decrease the steady state error signal. Try to give the answer. To improve the stability of the control system and to decrease the steady state error which controller is useful. Option A, p i d controller. Option B, proportional controller. Option C, p i controller. Option D, p d controller. The correct answer is the p i d controller. Advantages of p i d controller. It only allows the controller acts on the error between the desired signal and the controlled signal. Hence no extra measurement of the internal states are needed. The tuning can be done through trial and error or look up table. Not much knowledge of the plant is required for tuning. It is efficient and robust against some common uncertainties if properly tuned. Easy to implement in hardware through the filters and also easy to implement through microcontrollers and PLC. These are references. Thank you.